on Revetment Walls. 491 



arbitrary functions, whose form would have to be assigned so as 

 to make a certain quantity or the greatest of a set of quantities a 

 minimum ; but the peculiar manner in which the internal forces 

 are defined as subject to satisfy not an equation or system of 

 equations, but a law of inequality, must render it a task exceed- 

 ing the present powers, at all events,' of the writer of this paper, to 

 arrive at a result by the direct application of the Calculus referred 

 to. In order to pave the way to the discussion of the more 

 general inquiry, I shall commence with examining whether 

 under any and what circumstances the forced solution of Coulomb 

 and his followers, founded upon the notion of what have been (it 

 seems to me incautiously) termed planes of fracture or rupture 

 (but which really mean no more than planes for which friction at 

 each point thereof is acting with its utmost energy, i. e. if we 

 please so to say, planes of greatest frietional energy*), is the true 



* It is obvious that the notion of the planes in question being the planes 

 in which the earth would begin with crumbling, if the equilibrium were 

 disturbed by the wall giving way (for such is the idea intended to be con- 

 veyed by their being called planes of rupture), is quite irrelevant to the 

 determination of their position, and to the solution of the question of the 

 thrust in the wall. But such a notion in itself is objectionable, as assuming 

 a physical fact for which there is no just ground. The idea, or rather I may 

 say the metaphysical process, which unconsciously has swayed Coulomb 

 and his followers to give them this name, appears to me to be the follow- 

 ing. " Since it is only along these planes that friction is acting at its full 

 energy, and since, when motion ensues, friction must be acting at its full 

 energy, therefore a change must have taken place in the friction of any 

 other plane before motion can take place along it, which change does 

 not take place along the planes in question. Now every change must 

 operate in time, therefore the motion must have begun along the planes 

 of greatest friction before it can have taken place along any other." But 

 it is a most dangerous proceeding, and fraught with errors familiar to 

 mathematicians, to attempt to reason from the conditions of equilibrium 

 to those of incipient motion ; and that dynamical considerations, and not 

 statical, must decide the incipient directions of the motion in the case before 

 us, will be obvious when we reflect that the friction might be supposed to 

 become nil, and then we should be treating of a perfect fluid, in which case 

 the planes of rupture disappear, but none the less would motion take place 

 in determinate directions on any wall of the reservoir containing the fluid 

 giving way. A notable example of the important distinction between rest 

 and equilibrium is afforded by the question (which, I am informed, ori- 

 ginated in Caius College, Cambridge) of finding the tension of a rope by 

 which a bucket full of water, with a cork tied to its bottom, is fastened to 

 a fixed point, at the moment when the fastening is cut or gives way. At 

 that moment the vertical pressure in the bottom of the bucket, supposing 

 the specific gravity of the cork to be one-fourth that of water, if it could 

 be estimated on statical principles, i. e. with reference to the elevation of 

 the surface of the fluid [and some non-mathematical physicists might 

 easily suppose it could be so estimated, since motion has not yet taken 

 place, but is only imminent], would be the weight of the bucket together 

 with that of the water, together with four times that of the cork, and so it 

 would appear as if the tension would be increased by the cutting of the 



2 K2 



